Snedecor and Cochran (1967, Section 6.18) presented data obtained in 1942 from South Dakota on the relationship between the size of farms (in acres) and the number of acres planted in corn. Summary statistics are presented in Table 12.14. Note that the sample standard deviations rather than the sample variances are given. In addition, the pooled standard deviation is 0.4526.

(a) Give the one-way analysis of variance model with all of its assumptions. Can any problems with the assumptions be identified?

(b) Give the analysis of variance table for these data. Test whether there are any differences in corn acreages due to the different sized farms. Use α = 0.01.

(c) Test for differences between all pairs of farm sizes using α = 0.01 tests.

(d) Find the sum of squares for the contrast defined by the following coefficients:

Table 12.14: Acreage in corn for different sized farms.
Farm Sample Sample Sample acres size mean std. dev.
80 5 2.9957 0.4333 160 5 3.6282 0.4056 240 5 4.1149 0.4169 320 5 4.0904 0.4688 400 5 4.4030 0.5277 Table 12.15: Weights (in pounds) for various heights (in inches).
Sample Sample Sample Height size mean variance 63 3 121.6¯6 158.33¯3 65 4 131.25 72.91¯3 66 2 142.50 112.500 72 3 171.6¯6 158.33¯3 Farm 80 160 240 320 400 Coeff. −2 −1 0 1 2 What percentage is this of the treatment sum of squares?

(e) Give 95% confidence and prediction intervals for the number of acres in corn for each farm size.